They choose some fairly weird numbers to say "this is why AV is not much good".
For instance, they imagine a candidate C whose first preference voters divide up their second preference 'honestly' as:
A 600
B 6000
no 2nd preference: 6000
Now, you would think that means that B and C have a fair amount in common, but not A. But then, to try to claim that tactical voting might still happen, they ask us to accept that B's first preference voters 'honestly' divide their 2nd preference as:
A 12750
C 0
no 2nd preference: 0
which implies that voters think A and B have a lot in common, and C is totally different. But they have to come up with an unrealistic division of votes like this to claim that A could ask some supporters to vote tactically in the 1st round. In reality, tactical voting is a risky thing to attempt in AV - it's only worth doing if there's a 3rd party you think you can promote above the 2nd party that you're afraid of, but that means the 3rd party already has to be almost as popular as the 2nd party, for it to work. And, since the 2nd party was almost as popular as your own party (otherwise there would be no point in doing all this), you've just created a new opponent for yourself, with a few more first preference votes than your old opponent had - which have come from your own total.
Worst of all, that site, although saying elsewhere
they want STV, not AV, then links, in their 'AV and tactical voting' page, to a
Michael White article about Ireland as an answer to 'is this pure fantasy'? But what White describes is "disciplined vote management"
in an STV system. It's not the scenario they had put forward at all; it's a description of how a powerful party gets as many seats as possible in a multi-member constituency. It's asking Fine Gael supporters to vary which Fine Gael candidate they put as first choice - which is 'honest' voting for someone who thinks the party is the important thing, rather than an individual candidate. But in no way is it an argument against AV.